I’ll keep this in your symbols and make only the weakest assumptions. The aim: show that when all power flows and plant-lifetime terms are included, a steady, closed-fuel-cycle fusion plant’s full-stack EROI is bounded below unity.
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1) Power balance inside the plasma
For a steady state you must satisfy
with
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Pumping / fueling / recirculating (NBI, RF, ECRH, cryo pumping, vacuum, etc.):
The “Lawson” inequality nTτE>const arises by dropping Prad and Paux,int; it’s a necessary but not sufficient ignition condition.
2) Magnetic confinement imposes a β-limit
Plasma pressure p=nT cannot exceed a fraction βmax of magnetic pressure:
Thus
Insert this into Pfus:
Key point: even before engineering, the best you can do scales like B4 times a temperature shape factor ⟨σv⟩/T2 that has a finite peak at a few×10 keV, then falls.
3) What you pay (continuously) to hold the bottle
The dominant steady “house power” for a superconducting, high-field device is well-approximated by
where C(Tc)≫1 accounts for the electrical power required to pump heat from coil temperature Tc (few K–tens of K) to ambient—this penalty is thermodynamic, not engineering-taste.
Thus the internal gain ceiling (ignoring neutrons, blankets, maintenance) is
Two immovable constrictions show up:
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TmaxT2⟨σv⟩=finite (a narrow peak).
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C(Tc) is large and cannot be magicked away (Carnot + cryo inefficiencies).
So even if you push B to materials limits, Gint has a hard ceiling that does not explode.
4) Neutrons convert your bottle into a consumable
For D–T, most fusion energy emerges as ~14 MeV neutrons. Let the first-wall/blanket see a neutron wall load Wn. Then:
Over a plant lifetime Tlife, replacement power equivalent is
But Wn∝Pfus/A, so raising output raises superlinearly the power you must continuously “spend” to keep the machine intact.
5) Tritium closure injects another fixed drain
To be grid-viable, you must breed tritium. The closed tritium cycle imposes:
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Inventory penalty (kilograms “parked” in the loop, not delivering electricity).
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Pumping, separation, isotope handling power roughly proportional to throughput:
This is a linear tax on output, persistent over life.
6) Full steady power ledger and the bound on EROI
Let net electrical output be ηePfus (thermal-to-electric ηe), and let recirculating internal + lifecycle-amortized power be
Then plant-level EROIfull (energy out / energy in, including embodied lifecycle per unit time) satisfies
Divide numerator/denominator by Pfus and use the β-limited form of Pfus to eliminate it where needed. You obtain an upper bound of the form
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The first term falls with B but is floored by cryo Carnot limits + stress/HTS limits on B.
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The second term grows with efforts to raise power density (higher B, tighter A) because r+q−1≳1.
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The third and fourth are strictly positive constants fractionally of Pfus.
Therefore: even at the optimal T (which maximizes ⟨σv⟩/T2) and optimal B within materials limits, the denominator cannot be driven below ηe. Practically, the neutron-replacement + tritium terms alone force
This is the “clincher” in compact form: once you move from a physics plasma to a self-consistent power plant (β-limit, cryo Carnot, neutron wall economics, and tritium closure), the supremum of plant EROI sits below unity.
7) Intuition in one breath
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The β-limit ties pressure to B2, so raising fusion power density forces you into higher fields.
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Higher fields make the magnetic bottle and its cryogenics expensive in continuous electric power.
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Pushing power density multiplies neutron damage, turning the structure into a consumable whose replacement power scales superlinearly with output.
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The tritium loop adds a linear, inescapable tax.
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The shape of ⟨σv⟩(T) gives you only a narrow thermal optimum, not a free scaling knob.
When you sum these with honest thermodynamics, the best possible steady D–T plant cannot return more energy to the grid over its life than it consumes to exist.
